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Tagged with generating-functions stirling-numbers
4 questions
16
votes
1
answer
584
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What is this sequence?
This is again a question that I asked at Stack Exchange, but got no answer so far, so I am trying here.
Let:
$$ a_n=\sum_{k\ge0}(k+1) {n+2\brack k+2}(n+2)^kB_k$$
$B_k$ is the Bernoulli number. ${n\...
- 505
4
votes
1
answer
407
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Inverse relationship between Stirling numbers of the first and second kind via generating functions
In combinatorics, a well-known result is that the matrix formed by the Stirling numbers of the second kind $\left(S(n,k)\right)_{n,k\geq 0}$ and the matrix of the signed Stirling numbers of the first ...
- 151
3
votes
2
answers
463
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Ask for a reference or a proof of a combinatorial identity $\sum_{k=0}^n\binom{2n+1}{2k}\binom {k}{m} =2^{2(n-m)}\frac{2n+1}{2(n-m)+1}\binom{2n-m}{m}$
Could you please recommend a reference to or supply a proof of the following identity \eqref{combin-ID-Maclaurin}, or \eqref{first-equiv-form}, or \eqref{combin-ID-Mac-Equiv}, or \eqref{combin-ID-Mac-...
- 1,101
1
vote
1
answer
921
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What is the inverse Laplace transform of $\frac{(1/s)_{n}}{s} $?
Introduction
So far, I have found (p. 5) the following generating functions of the unsigned Stirling numbers of the first kind:
\begin{equation} \tag{1} \label{1} \sum_{l=1}^{n} |S_{1}(n,l)|z^{l} = (z)...
- 4,829